Page:Elementary Principles in Statistical Mechanics (1902).djvu/180

We define a path as the series of phases through which a system passes in the course of time when the external coördinates have fixed values. When the external coördinates are varied, paths are changed. The path of a phase is the path to which that phase belongs. With reference to any ensemble of systems we shall denote by ${\overline {D}}|_{p}$ the average value of the density-in-phase in a path. This implies that we have a measure for comparing different portions of the path. We shall suppose the time required to traverse any portion of a path to be its measure for the purpose of determining this average.

With this understanding, let us suppose that a certain ensemble is in statistical equilibrium. In every element of extension-in-phase, therefore, the density-in-phase $D$ is equal to its path-average ${\overline {D}}|_{p}$. Let a sudden small change be made in the external coördinates. The statistical equilibrium will be disturbed and we shall no longer have $D={\overline {D}}|_{p}$ everywhere. This is not because $D$ is changed, but because ${\overline {D}}|_{p}$ is changed, the paths being changed. It is evident that if $D>{\overline {D}}|_{p}$ in a part of a path, we shall have $D<{\overline {D}}|_{p}$ in other parts of the same path.

Now, if we should imagine a further change in the external coördinates of the same kind, we should expect it to produce an effect of the same kind. But the manner in which the second effect will be superposed on the first will be different, according as it occurs immediately after the first change or after an interval of time. If it occurs immediately after the first change, then in any element of phase in which the first change produced a positive value of $D-{\overline {D}}|_{p}$ the second change will add a positive value to the first positive value, and where $D-{\overline {D}}|_{p}$ was negative, the second change will add a negative value to the first negative value.

But if we wait a sufficient time before making the second change in the external coördinates, so that systems have passed from elements of phase in which $D-{\overline {D}}|_{p}$ was originally positive to elements in which it was originally negative, and vice versa, (the systems carrying with them the values